Communications in Mathematical Sciences

Volume 16 (2018)

Number 1

A note on the stability of implicit-explicit flux-splittings for stiff systems of hyperbolic conservation laws

Pages: 1 – 15



Hamed Zakerzadeh (Institut für Geometrie und Praktische Mathematik, RWTH Aachen University, Aachen, Germany)

Sebastian Noelle (Institut für Geometrie und Praktische Mathematik, RWTH Aachen University, Aachen, Germany)


We analyze the stability of implicit-explicit flux-splitting schemes for stiff systems of conservation laws. In particular, we study the modified equation of the corresponding linearized systems. We first prove that symmetric splittings are stable, uniformly in the singular parameter $\varepsilon$. Then, we study non-symmetric splittings. We prove that for the isentropic Euler equations, the Degond–Tang splitting [Degond & Tang, Comm. Comp. Phys., 10:1–31, 2011] and the Haack–Jin–Liu splitting [Haack, Jin Liu, Comm. Comp. Phys., 12:955–980, 2012], and for the shallow water equations the recent RS-IMEX splitting are strictly stable in the sense of Majda–Pego. For the full Euler equations, we find a small instability region for a flux splitting introduced by Klein [Klein, J. Comp. Phys., 121:213–237, 1995], if this splitting is combined with an IMEX scheme as in [Noelle, Bispen, Arun, Lukáčová, Munz, SIAM J. Sci. Comp., 36:B989–B1024, 2014].


stiff hyperbolic systems, flux-splitting, IMEX scheme, asymptotic preserving (AP) property, modified equation, stability analysis

2010 Mathematics Subject Classification

35L65, 65M08, 76M45

The first author was supported by the scholarship of RWTH Aachen University through Graduiertenförderung nach Richtlinien zur Förderung des wissenschaftlichen Nachwuchses (RFwN).

Received 28 July 2017

Accepted 23 August 2017

Published 29 March 2018